Systematic Vertex Corrections through Iterative Solution of Hedin's Equations Beyond theGWApproximation

We present a general procedure for obtaining progressively more accurate functional expressions for the electron self-energy by iterative solution of Hedin's coupled equations. The iterative process starting from Hartree theory, which gives rise to the $\mathrm{GW}$ approximation, is continued further, and an explicit formula for the vertex function from the second full cycle is given. Calculated excitation energies for a Hubbard Hamiltonian demonstrate the convergence of the iterative process and provide further strong justification for the $\mathrm{GW}$ approximation.